Abstract
The integration of converter-interfaced generators (CIGs) into power systems is rapidly replacing traditional synchronous machines. To ensure the security of power supply, modern power systems require the application of grid-forming technologies. This study presents a systematic small-signal analysis procedure to assess the synchronization stability of grid-forming virtual synchronous generators (VSGs) considering the power system characteristics. Specifically, this procedure offers guidance in tuning controller gains to enhance stability. It is applied to six different grid-forming VSGs and experimentally tested to validate the theoretical analysis. This study concludes with key findings and a discussion on the suitability of the analyzed grid-forming VSGs based on the power system characteristics.
THE transition to a power system dominated by renewable energy sources (RESs) interfaced with power electronics converters is an indisputable fact. These new converter-interfaced generators (CIGs) have been installed across all voltage levels, from transmission to distribution networks [
Fortunately, new operation modes of CIGs have been developed to address the technical challenges associated with large-scale RES integration. Among these, the grid-forming control has gained prominence [
The OCLs comprise the active power control loops (APCLs) and reactive power control loops (RPCLs). The most common APCLs for grid synchronization are: ① droop control [
Regarding the ICL, various controllers are presented: ① open-loop controller [
The combination of different OCL and ICL blocks results in a plethora of grid-forming controllers of CIGs. Therefore, it is required to assess the performance of these controllers systematically considering the following aspects [
However, recent studies suggest that the SCR may not provide accurate estimations of system impedance or strength in converter-dominated power systems. This is mainly because the short-circuit current contributions from voltage source converters (VSCs) and synchronous generators are different. Moreover, it is uncertain how to effectively utilize the SCR information to design the CIG controllers [
Considering this background, the specific contributions of this study can be summarized as follows.
1) The stability ranges are identified for different types of grid-forming VSGs based on the value SCR and ratio R/X of the grid through a small-signal analysis.
2) A systematic procedure is proposed based on the analysis of participation factors, which extends the operating range of grid-forming VSGs by identifying the control state variables with the major weight on the oscillation modes causing system instability.
3) The control loops that cause synchronization instability for each type of grid-forming VSG are identified. This will show that the ICL dynamics of grid-forming VSGs cannot be neglected in the stability analysis.
4) The proposed systematic procedure for improving the synchronization stability of grid-forming VSGs is experimentally validated.
The remainder of this paper is organized as follows. Section II presents the state-space modeling of grid-forming VSGs. Section III outlines the systematic small-signal analysis procedure to evaluate the effects of SCR and R/X on the stability range, which is applied to a case study. From this analysis, the controller gains are re-tuned to extend the stability range for networks with different characteristics. Section IV describes the experimental validation. Section V discusses considerations and important properties of grid-forming VSGs that should be additionally considered beyond the findings obtained from the small-signal analysis. Section VI concludes by describing the main findings and gives an assessment of the most suitable OCL-ICL combinations based on power system characteristics.
This section details the state-space model of a grid-connected VSC and the most common OCLs and ICLs to implement a grid-forming VSG. The differential equations of the grid-forming VSGs are modeled in the frame, synchronized with the VSG angle . The models of the APCL and RPCL are nonlinear. Consequently, the linearization of the state-space model is required prior to implementing the systematic small-signal analysis procedure proposed in Section III.
The CIG considered in this study is a three-phase three-wire VSC connected to the grid via an LCL coupling filter, which includes a series damping resistor alongside the capacitor, as illustrated in

Fig. 1 One-line diagram of a three-phase three-wire grid-connected VSC.
The differential equations for the grid-connected VSC in the dq frame are given as:
(1) |
(2) |
(3) |
where is the VSC voltage; is the voltage of RC branch; is the grid-side inductor voltage; is the VSC-side current; is the grid-side current; and is the capacitor current; is the angular speed of VSG; and are the VSC and grid-side inductances, which are along with inner resistances and , respectively; and and are the damping resistance and capacitance of the coupling filter, respectively.
The power grid is modeled based on its Thevenin equivalent as:
(4) |
where is the grid voltage; and and are the grid inductance and resistance, respectively.
The state-space model of the grid-connected VSC is represented in (1)-(4) and contains six state variables:
(5) |
Regarding the OCL, the APCL provides the angular speed and angle of virtual rotor, i.e., and , respectively, whereas the RPCL provides the virtual electromotive force, i.e., . The OCL control schemes, comprising two possible implementations based on S-VSG [

Fig. 2 Diagrams of OCL and ICL blocks considered for analyzed grid-forming VSGs. (a) OCL. (b) ICL.
Both S-VSG and PI-VSG consider that the grid voltage rotates at the nominal frequency , which acts as a perturbation with no influence on the system dynamics [
(6) |
The OCL formulation based on the S-VSG or PI-VSG is presented as follows.
The APCL based on S-VSG can be expressed as:
(7) |
where is the measured active power; is the active power reference, i.e., virtual mechanical power; is the rotational inertia; and is the damping coefficient.
Combining (6) and (7) leads to:
(8) |
The RPCL consists of a PI controller, which can be formulated as:
(9) |
where is the integral of reactive power error; is the measured reactive power; is the reactive power reference; is the rated electromotive force; and and are the proportional and integral gains of PI controller in the RPCL, respectively.
Therefore, the S-VSG introduces three state variables in the state-space model:
(10) |
The APCL based on PI-VSG, consisting of a PI controller applied to the active power error, can be written as:
(11) |
where is the integral of active power error; and and are the proportional and integral gains of PI controller in the APCL, respectively.
The RPCL of PI-VSG is identical to that of the S-VSG. Therefore, the PI-VSG introduces three state variables in the state-space model:
(12) |
Three types of ICLs are considered for the grid-forming VSG: ① voltage controller by means of cascaded controller [
A cascaded voltage controller composed of voltage and current control loops is used to control the capacitor voltage of the LCL filter. The voltage reference of RC branch is computed as [
(13) |
where is the virtual impedance; and is the virtual electromotive force computed by the RPCL.
A PI voltage controller provides the setpoints to the current control loop:
(14) |
where is the integral of the voltage error; is the VSC-side current reference; and and are the proportional and integral gains of the PI voltage controller in VC-ICL, respectively.
The tracking of is carried out in the current control loop, which is based on a PI current controller and modeled as:
(15) |
where is the integral of the current error; and and are the proportional and integral gains of the PI current controller in VC-ICL, respectively.
Based on (13)-(15), the VC-ICL adds four new state variables to the state-space model:
(16) |
CC-ICL controls the grid-side inductor current of LCL filter. To achieve this, a PI controller is applied to the current error. The grid-side current reference is obtained using a virtual admittance and a low-pass filter (LPF) as:
(17) |
(18) |
where is the time constant of LPF; is the input current of LPF; and is the virtual admittance matrix, with and being the virtual conductance and susceptance, respectively. The differential equations of the PI current controller, including the cross-coupling cancellation and feed-forward terms, are expressed as:
(19) |
where ; and and are the proportional and integral gains of the PI current controller in CC-ICL, respectively.
Therefore, based on (17)-(19), the CC-ICL adds four state variables to the state-space model:
(20) |
The OL-ICL directly applies the virtual electromotive force to the VSC terminals. A slight modification of is proposed using a transient virtual resistor (TVR) to provide more damped current dynamics [
(21) |
where is the output voltage of TVR; and and are the gain and cut-off frequency of high-pass filter, respectively. Then, the VSC terminal voltage is computed as:
(22) |
The OL-ICL introduces two new state variables to the state-space model:
(23) |
It is important to note that for different ICLs, the power terms used in the corresponding OCL blocks are computed at different nodes, i.e., at the capacitor branch of LCL filter for the VC-ICL, at the POI for the CC-ICL, and at the VSC terminals for the OL-ICL.
Therefore, six state-space models of grid-forming VSG are derived based on the selected OCL and ICL blocks, which can be formulated in a generalized form as:
(24) |
where A and B are the state-space matrices; and the state vector is composed of the state variables of grid-connected VSC in (5), the state variables of OCL in (10) or (12), and state variables of ICL in (16), (20), or (23).
This section presents a systematic small-signal analysis procedure to ensure the synchronization stability and good dynamic performance of the grid-forming VSGs in power systems with different SCR and .
This subsection presents the steps of the proposed systematic procedure for improving the synchronization stability of grid-forming VSGs. The fundamentals of the small-signal analysis are presented in Supplementary Material A. The proposed systematic procedure consists of the following steps.
Step 1: validate the linear model against the original nonlinear model by comparing the dynamic performance of some key magnitudes, such as the active and reactive power used in the OCL.
Step 2: identify stable and unstable regions over a wide range of SCR and R/X through a small-signal analysis based on the linear model.
Step 3: identify the eigenvalues associated with the critical oscillation modes at the boundary between stable and unstable regions. These eigenvalues are those that lead the system to unstable operation.
Step 4: based on these critical oscillation modes, detect the critical state variables by analyzing their corresponding participation factors for the unstable region identified in Step 3. In particular, the critical state variables are those with the largest participation factors in the critical oscillation modes. Detailed information on the computation and interpretation of participation factors can be found in Supplementary Material A.
Step 5: perform a sensitivity analysis of the controller gains associated with the critical state variables within the unstable region identified in Step 3 to increase the synchronization stability range of the grid-forming VSG.
This subsection presents a case study using a CIG with the parameters shown in
Parameter | Value | Parameter | Value |
---|---|---|---|
(mH) | 1.25 | () | 0.08 |
() | 0.04 | () | 0.0016 |
(mH) | 1.25 | () | 47.36 |
() | 0.04 | (V/W) | 0.0163 |
() | 0 | (rad/s) | 60 |
() | 0.0012 | (ms) | 1.6 |
() | 2.03 | (V/A) | 1.25 |
(V/W) | 0.0016 | (V/A) | 40 |
() | 0.09 | (S) | 0 |
(rad/s) | 100 | (S) | 1.25 |
() | 10 | (A/V) | 0.1 |
() | 4 | (A/V) | 0.1 |
(V) | (V/A) | 0.1 | |
(V) | (V/A) | 15.1 |
Step 1: the linear model is validated by comparing its step response with that of the nonlinear model.

Fig. 3 Step response of linear and nonlinear models to active and reactive power step changes of kW and kvar. (a) S-VSG. (b) PI-VSG.
The active power and reactive power are computed at the midpoint of the LCL filter to which the capacitor is connected. The active and reactive power step changes are applied to all the combinations of OCL and ICL blocks at different time instants in a compact manner without overlaps. In addition, note that the steady-state active power and reactive power for each ICL are different, since the power involved in the OCL depends on the implemented ICL, as previously described in Section II.
Step 2: the objective of this step is to evaluate the stability of the six grid-forming VSGs with different combinations of OCL and ICL blocks in different power grids. According to the implemented OCL and ICL blocks, the six grid-forming VSGs can be termed as S-VC, S-CC, S-OL, PI-VC, PI-CC, and PI-OL. A wide range of SCR and R/X is used to characterize different power grids. According to the standard IEEE Std. 1204-1997 [

Fig. 4 Stable region for six grid-forming VSGs within and . (a) S-VC. (b) S-CC. (c) S-OL. (d) PI-VC. (e) PI-CC. (f) PI-OL.
The results for the PI-VSG show that the ICLs based on voltage control, i.e., PI-OL and PI-VC, are generally stable for LV grids, i.e., with high R/X, except for very high SCR. This stability is compromised when the power grid becomes inductive for , regardless of SCR. In addition, it is observed that both PI-OL and PI-VC are unstable when regardless of R/X. This means that PI-OL and PI-VC are not recommended for CIGs connected to grids with low impedance and/or high reactance, i.e., HV grids. Regarding the PI-CC, the system is stable with medium/high SCR and medium/low R/X, i.e., HV and MV grids, achieving full stability for regardless of R/X. In addition, a large R/X contributes to the system stability, irrespective of SCR. This suggests that PI-CC is not suitable for networks with high impedance and/or high reactance. In the case of the S-VSG, the ICLs based on voltage control, i.e., S-VC and S-OL, follow a similar pattern to PI-VC and PI-OL. However, they guarantee stability across the entire range of SCR and R/X. This does not happen in the S-CC, whose performance is very similar to the PI-CC, but achieving a wider stable region for regardless of R/X. This analysis derives that the performance is similar among the following groups of grid-forming VSGs: ① PI-OL and PI-VC, ② PI-CC and S-CC, and ③ S-OL and S-VC. In addition to the information provided by this small-signal stability analysis, it is possible to provide some physical insights about the performance of these groups. PI-OL and PI-VC perform like voltage sources. Accordingly, their stability deteriorates in stiff networks with high SCR, as two voltage sources coupled via a small impedance may experience significant active power variations due to minor phase differences. In this regard, increasing the coupling impedance, i.e., reducing SCR, the stable region of the system widens. By contrast, PI-CC and S-CC behave like a current source from the view of synchronization. They may perform well when connected to stiff grids but deteriorate in weak grids with low SCR. The performances of S-OL and S-VC are similar to their counterparts based on the PI-VSG, i.e., PI-OL and PI-VC, respectively. However, due to the selected damping coefficient , they offer a fully stable operation within the analyzed intervals of SCR and R/X. For this reason, no further steps of the proposed procedure are applied to these grid-forming VSGs.
Step 3: according to
VSG | SCR | R/X | Eigenvalue | Re() | (Hz) | |
---|---|---|---|---|---|---|
PI-OL | 25.00 | 0.32 | , | 14.92 | 67.21 | |
PI-VC | 25.00 | 0.32 | , | 27.80 | 97.59 | |
PI-CC | 6.25 | 0.32 | , | 5.40 | .00 | |
S-CC | 6.25 | 0.32 | , | .00 |
Step 4: once the critical oscillation modes of each grid-forming VSG have been identified, an in-depth analysis is performed within the boundary between the stable and unstable regions.

Fig. 5 Evolution of eigenvalues associated with critical oscillation mode of PI-OL and PI-CC when SCR and R/X change. (a) PI-OL. (b)PI-CC.
x | Participation factor | |||||||
---|---|---|---|---|---|---|---|---|
PI-OL (, ) | PI-VC (, ) | PI-CC (, ) | S-CC (, ) | |||||
, (stable) | , (unstable) | , (stable) | , (unstable) | , (stable) | , (unstable) | , (stable) | , (unstable) | |
0.03 | 0.18 | 0.03 | 0.23 | 0 | 0 | 0 | 0 | |
0 | 0 | |||||||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0.01 | 0 | 0 | 0 | 0 | |
0 | 0 | |||||||
0 | 0 | |||||||
0 | 0.18 | 0 | 0.18 | |||||
0 | 0.18 | 0 | 0.18 | |||||
0 | 0 | 0 | 0 | 0 | 0 | |||
0 | 0 | 0 | 0 | 0 | 0 | |||
0.01 | 0.03 | |||||||
0.01 | 0.03 |
Similarly, the sensitivity analysis of the critical oscillation mode of the PI-CC associated with the complex eigenvalues and is shown in
Step 5: the aim of this step is to increase the stability range and improve the dynamic performance of grid-forming VSGs affected by the network impedance variation. This is accomplished by re-tuning the controller gains based on the small-signal analysis.
The stability problems of the PI-OL and PI-VC are related to the APCL in OCL; therefore, according to (11), re-tuning their PI controller gains is logical. The proportional gain is modified, as the integral gain is directly related to the inertia constant, and cannot be modified so that the VSG inertial response remains unaffected.

Fig. 6 Evolution of eigenvalues for PI-OL and PI-VC when decreases. (a) PI-OL. (b) PI-VC.
The instability of S-CC and PI-CC is fundamentally different because the root of this problem lies within the virtual admittance and its LPF. Therefore, the targeted control parameters to be re-tuned are , , and . Note that increasing destabilizes the system, whereas decreasing and/or increasing have a stabilizing effect [

Fig. 7 Evolution of eigenvalues for PI-CC. (a) decreases. (b) increases.
These re-tuned control parameters significantly extend the stable region for all grid-forming VSGs, as shown in

Fig. 8 Stable region with re-tuned control parameters. (a) S-VC. (b) S-CC. (c) S-OL. (d) PI-VC. (e) PI-CC. (f) PI-OL.
This section aims to experimentally validate the stability and dynamic response of the grid-forming VSGs to sudden changes in grid impedance.
The experimental validation of the six grid-forming VSGs is conducted using a testbed based on the single-line diagram shown in

Fig. 9 Experimental testbed. (a) One-line diagram. (b) VSC and LCL filter. (c) Feeder.
The experimental tests evaluate the stability of different grid-forming VSGs against variations in the grid impedance. Initially, in scenario , the contactor N1 is closed and N2 is open, and the power system is characterized by and . At a given time, the contactor N2 is closed, connecting both feeders in parallel, namely scenario . This drastically changes the power system characteristics to and . The initial and final SCR and R/X for experimental test are marked in Figs.
The evolution of the active and reactive power of each grid-forming VSG is presented in

Fig. 10 Evolution of active and reactive power of each grid-forming VSG. (a) S-VC. (b) S-CC. (c) S-OL. (d) PI-VC. (e) PI-CC. (f) PI-OL.

Fig. 11 Evolution of injected current of each grid-forming VSG. (a) S-VC. (b) S-CC. (c) S-OL. (d) PI-VC. (e) PI-CC. (f) PI-OL.
Then, the contactor N2 is closed at s (scenario ), resulting in a transient response in each grid-forming VSG before they return to their corresponding setpoints. First, the analyzed grid-forming VSGs are stable, as indicated in the theoretical analysis, because the change of power system characteristics is within the identified stable region depicted in
This experimental performance aligns with the small-signal analysis results, as presented in
VSG | Mode | Scenario A | Scenario B | ||
---|---|---|---|---|---|
(Hz) | (Hz) | ||||
S-VC | , | 57.5 | 58.8 | ||
S-CC | , | 2084.6 | 2668.1 | ||
S-OL | , | 64.6 | 68.7 | ||
PI-VC | , | 55.8 | 59.0 | ||
PI-CC | , | 2084.9 | 2668.5 | ||
PI-OL | , | 63.4 | 68.3 |
This section discusses some other features, in addition to synchronization stability, of the analyzed grid-forming VSGs that must be considered. These features depend on the implemented OCL and ICL blocks and their corresponding parameterization. In this regard, the fact that the initial settings of the controllers analyzed in this study are chosen according to the state of the art must be highlighted.
Regarding the OCL, the S-VSG leads to wider stability areas compared with PI-VSG, even the PI-VSG gains are tuned, as shown in
First, the damping and PFR in the S-VSG are coupled, making it impossible to define these two design parameters separately without introducing additional modifications into the APCL. Thus, a large damping coefficient , which is required because of the POI characteristics, may lead to a large PFR. This in turn may produce a CIG overload in the event of a frequency disturbance. By contrast, the damping and PFR of the PI-VSG are fully decoupled and thus can be set independently. Second, due to the larger damping, the dynamic response of the S-VSG to a reference step change is slower than that of the PI-VSG, as shown in
Regarding the ICLs, they must be considered to assess the synchronization stability of grid-forming VSGs, as evidenced in the conducted small-signal analysis. The analytical results reveal that the grid-forming VSGs adopting OL-ICL exhibit a superior performance, regardless of the selected OCL, as shown in
Finally,
VSG | Stability | HV grid | MV grid | LV grid | Property |
---|---|---|---|---|---|
S-OL | Well-damped oscillation modes |
+Inertial response Coupled damping and PFR Lack of voltage or current control | |||
S-VC | Well-damped oscillation modes |
+Inertial response Coupled damping and PFR +Voltage control +Inverter current control | |||
S-CC |
1) No low-frequency critical oscillation modes 2) Poorly-damped high-frequency oscillation modes with low SCR 3) Robustness with high SCR | × |
+Inertial response Coupled damping and PFR +PQ control at the POI +Grid current control | ||
PI-OL |
1) Well-damped oscillation modes 2) Robustness with low SCR |
+Inertial response +Decoupled damping and PFR +Faster time response Lack of voltage or current control | |||
PI-VC |
1) Poorly-damped low-frequency oscillation mode with high SCR and low R/X 2) Robustness with low SCR | × |
+Inertial response +Decoupled damping and PFR +Faster time response +Voltage control +Inverter current control | ||
PI-CC |
1) No low-frequency critical oscillation mode 2) Poorly-damped high-frequency oscillation modes with low SCR 3) Robustness with high SCR | × |
+Inertial response +Decoupled damping and PFR +Faster time response +PQ control at the POI +Grid current control |
Note: the symbol represents that the grid-forming VSG is applicable to HV, MV, or LV grid; the symbol × represents that the grid-forming VSG is not applicable to HV, MV, or LV grid; the symbol + represents the advantages; and the symbol represents the disadvantages.
This study proposes a systematic small-signal analysis procedure to evaluate and improve the synchronization stability of grid-forming VSGs. The proposed systematic procedure is applied to six grid-forming VSGs, derived using different combinations of OCL and ICL blocks. Given that the synchronization stability is extremely sensitive to power system characteristics, the analysis considers a wide range of SCR and R/X to draw general conclusions.
The proposed systematic procedure enhances the synchronization stability of grid-forming VSGs, providing insights into how controller gains should be re-tuned for this purpose. The application of the proposed systematic procedure reveals that the S-VSG, parameterized according to state-of-the-art practices, has a broader stable region as compared with PI-VSG. This is caused by a large damping coefficient, which is required because of the POI characteristics. This parameter may lead to a slower transient response and a larger PFR, which could cause a CIG overload in the event of a frequency disturbance. In addition, the ICLs play a crucial role in the stability of grid-forming VSGs. For the ICLs based on voltage control, i.e., VC-ICL and OC-ICL, the stability is determined by low-frequency oscillation modes associated with the APCL. By contrast, the stability of grid-forming VSGs with ICLs based on current control, i.e., CC-ICL, is linked to high-frequency oscillation modes associated with the current controller. Therefore, the synchronization stability of grid-forming VSGs does not solely depend on low-frequency modes as traditionally defined, as high-frequency modes can also emerge due to the rapid control actions within the ICLs. Finally, VC-ICL and OC-ICL are more suitable for networks with medium/low SCR, whereas CC-ICL is more effective for networks with medium/high SCR. In general, a high R/X improves stability across all grid-forming VSGs, as expected.
Future research will focus on extending this systematic analysis to other aspects of grid-forming VSGs, including power system dynamics and unbalanced operation.
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